Advancements in quantum physics have facilitated understanding and enhancement of existing technologies. For example, understanding the quantum behavior of electrons in semiconductor materials has made it possible to develop and produce smaller and faster semiconducting devices. Recent and promising advancements in quantum physics are now being exploited to produce new technologies. For example, certain quantum systems can be used to encode and transmit information in new ways. In addition, quantum systems can be used in optical lithographic and measuring devices to provide resolution and sensitivity limits that exceed those of non-quantum-based lithographic instruments and measuring devices.
Quantum systems comprising just two discrete states, represented by “|0” and “|1,” are often employed in a variety of quantum-system-based applications including quantum information encoding and processing, optical quantum lithography, and metrology. Examples of two-state quantum systems include any two photon states of an electromagnetic field, vertically and horizontally polarized photons, and the two spin states of an electron. A quantum system with two discrete states is called a “qubit system,” and the states |0 and |1, called “qubit basis states,” can also be represented in set notation as {|0, |1}. A qubit system can exist in the state |0, the state |1, or in any of an infinite number of states that simultaneously comprise both |0 and |1. Any of the states that include both |0 and/or |1 can be represented mathematically as a linear superposition of states:|ψ=α|0+β|1The state |ψ is called a “qubit,” and the parameters α and β are complex-valued coefficients satisfying the condition:|α|2+|β|2=1
In general, the sum of the square modulus of the coefficients is “1.” Performing a measurement on a quantum system collapses the state of the quantum system onto a basis state and produces an associated real-valued quantity. Performing a measurement on a quantum system is mathematically equivalent to projecting the state of the quantum system onto one of the basis states, and, in general, the probability of projecting the state of the quantum system onto a basis state is equal to the square of the coefficient associated with the basis state. For example, when the state |ψ of the qubit system is measured in the basis {|0,|1}, one has a probability |α|2 of finding the quantum system in the state |0 and a probability |β|2 of finding the quantum system in the state |1.
The infinite number of states associated with a qubit system can be geometrically represented by a unit-radius, three-dimensional sphere called a “Bloch sphere”:
          ψ    〉    =            cos      ⁢                          ⁢              (                  θ          2                )            ⁢                      0        〉              +                  ⅇ                  ⅈ          ⁢                                          ⁢          ϕ                    ⁢              sin        ⁡                  (                      θ            2                    )                    ⁢                      1        〉            where
0≦θ<π, and
0≦φ<2π.
FIGS. 1A-1C illustrate a Bloch sphere representation of a qubit system. In FIG. 1A, lines 101-103 are orthogonal x, y, and z Cartesian coordinate axes, respectively, and a Bloch sphere 106 is centered at the origin. There are an infinite number of points on the Bloch sphere 106, each point representing a unique state of a qubit system. For example, a point 108 on the Bloch sphere 106 represents a unique state of a qubit system that simultaneously comprises, in part, the state |0 and, in part, the state |1. However, once the state of the qubit system is measured in the basis {|0,|0}, the state of the qubit system is projected onto the state |0 110, in FIG. 1B, or onto the state |1 112, in FIG. 1C.
The state of a combined qubit system comprising two or more qubit systems is represented by a product of qubits, each qubit associated with one of the qubit systems. For example, consider a combined qubit system comprising a first qubit system and a second qubit system that is represented by the state:|ψ12=|ψ1|2 where the state of the first qubit system is:
                  ψ      〉        1    =            1              2              ⁢          (                                                0            〉                    1                +                                          1            〉                    1                    )      and the state of the second qubit system is:
                  ψ      〉        2    =            1              2              ⁢          (                                                0            〉                    2                +                                          1            〉                    2                    )      The state |ψ2 can also be written as a linear superposition of states:
                                                      ψ            〉                    12                =                                                          ψ              〉                        1                    ⁢                                                  ψ              〉                        2                                                  =                              1            2                    ⁢                      (                                                                                                    0                    〉                                    1                                ⁢                                                                          0                    〉                                    2                                            +                                                                                        0                    〉                                    1                                ⁢                                                                          1                    〉                                    2                                            +                                                                                        1                    〉                                    1                                ⁢                                                                          0                    〉                                    2                                            +                                                                                        1                    〉                                    1                                ⁢                                                                          1                    〉                                    2                                                      )                              where the terms |01, |02, |01|12, |111|02, and |11|12 are product states. Each product state in the state |ψ12 has an associated coefficient of 1/2, which indicates that when the state of the first qubit system is measured in the bases {|01,|11}, and the state of the second qubit system is measured in the basis {|02,|12}, there is a 1/4 probability of the combined qubit systems being found in any one of the product states. For example, when the states of the first and the second qubit systems are measured in the bases {|01,|11} and {|02,|12}, respectively, there is a 1/4 (|1/2|2) probability of projecting the state of the combined qubit system onto the product state |11|12.
The state of certain combined qubit systems cannot be represented by a product of associated qubits. These qubit systems are said to be “entangled.” Quantum entanglement is a property of quantum mechanics in which the states of two or more quantum systems are linked to one another, even though the quantum systems can be spatially separated. An example entangled state representation of an entangled two-qubit system is:
                  ϕ      〉        12    =            1              2              ⁢          (                                                                0              〉                        1                    ⁢                                                  1              〉                        2                          +                                                          1              〉                        1                    ⁢                                                  0              〉                        2                              )      
The entangled state |φ12 cannot be factored into a product of the qubits α1|01+β1|11 and α2|02+β2|12, for any choice of the parameters α1, β1, α2, and β2.
The state of an un-entangled, two-qubit system can be distinguished from the state of an entangled, two-qubit system as follows. Consider an un-entangled two-qubit system in the un-entangled state |ψ12. Suppose a measurement performed on the first qubit system in the basis {|01,|11} projects the state of the first qubit system onto the state |0. According to the state |ψ12, the state of the un-entangled, two-qubit system just after the measurement is the linear superposition of states (|01|02+|01|12)/√{square root over (2)}. When a second measurement is performed on the second qubit system in the basis {|02,|12} immediately following the first measurement in an identical reference frame, there is a 1/2 probability of projecting the state of the second qubit system onto the state |02 and a 1/2 probability of projecting the state of the second qubit system onto the state |12. In other words, the state of the second qubit system is not correlated with the state of the first qubit system. By contrast, consider an entangled, two-qubit system in the entangled state |φ12. Suppose that a first measurement performed on the first qubit system in the basis {|01,|11} also projects the state of the first qubit system onto the state |01. According to the entangled state |φ12, the state of the entangled, two-qubit system after the first measurement is the product state |01|12. When a second measurement is performed on the second qubit system in the basis {|02,|12}, the state of the second qubit system is |12 with certainty. In other words, the state of the first qubit system |01 is correlated with the state of the second qubit system |12.
Entangled quantum systems have a number of different and practical applications in disciplines such as metrology, cryptography, and many-qubit communications protocols. In metrology, for example, certain kinds of entangled quantum systems can be used to improve quantum-enhanced-phase measurements of a phase shift, φ, induced by an optical element on a beam of electromagnetic radiation. One conventional method for determining the phase shift φ of an optical element is to place the optical element within an internal, optical path of an interferometer. FIG. 2A illustrates an interferometer 200. The interferometer 200 comprises a first beamsplitter 202, a second beamsplitter 204, two mirrors 206 and 208, a first photon detector 210, and a second photon detector 212. The first beam splitter 202 is a 50:50 beam splitter that receives photons in input paths 214 and 216 and outputs photons in a first internal path, identified by directional arrows 218 and 220, and a second internal path, identified by directional arrows 222 and 224. The second beam splitter 204 is also a 50:50 beam splitter that recombines the photons transmitted by the first and second internal paths and outputs photons to the detectors 210 and 212. An optical element 226 with an unknown and sought after phase shift φ is located within the second internal path. The first beamsplitter 202 receives a single photon from either the input path 214 or the input path 216 and outputs a photon in the linear superposition of states:
          ψ    〉    =            1              2              ⁢          (                                  0          〉                +                            1          〉                    )      where
|0 is a state that represents a photon transmitted in the first internal path, and
|1 is a state that represents a photon transmitted in the second internal path. Just before the second beamsplitter 204, the optical element 226 causes a photon in the state |1 to acquire the phase shift φ, which places the qubit in the state:
          ψ    〉    =            1              2              ⁢          (                                  0          〉                +                              ⅇ                          ⅈ              ⁢                                                          ⁢              φ                                ⁢                                  1            〉                              )      The phase shift φ is determined from a quantum interference pattern that is detected by the detectors 210 and 212 and is built-up over a large number of trails. The state just before the second beamsplitter 204 for N trials is a product of N qubits and is given by:
                  ψ      〉        N    =            ∏              j        =        1            N        ⁢                  1                  2                      N            /            2                              ⁢              (                                                          0              〉                        j                    +                                    ⅇ                              ⅈ                ⁢                                                                  ⁢                φ                                      ⁢                                                          1                〉                            j                                      )            Photon interference at the second beamsplitter 204 produces over N trials a sinusoidal interference pattern of the form N cos(φ) that is detected by the detectors 210 and 212. The phase shift φ can be read from the interference pattern, and the uncertainty associated with an N trial determination of the phase shift φ, called the “shot-noise limit,” is given by:
      Δ    ⁢                  ⁢          φ      SL        =      1          N      
The expression for the shot noise limit is determined in accordance with estimation theory, as described in the book “Quantum Detection and Estimation Theory,” by C. W. Helstrom, Mathematics in Science and Engineering 123, Academic Press, New York, 1976. The shot noise limit is a value that represents the statistical dispersion, or standard deviation, in the experimentally determined value of the phase shift φ. Because the shot noise limit is inversely proportional to the square root of the number of trials, the uncertainty in the phase shift φ decreases as the number of trials N increases.
However, methods and systems that employ entangled qubit systems, such as an entangled N-qubit system in a “NOON” state, can be used to improve upon the shot noise limit by providing the optimal obtainable accuracy in the value of the phase shift φ. The N-qubit NOON state is a member of a larger class of maximally entangled qubits called “Greenberger-Horne-Zeilinger” (“GHZ”) states and is given by:
                  ψ      〉        NOON    =            1              2              ⁢          (                                              N            ,            0                    〉                +                                        0            ,            N                    〉                    )      where
|N,0 represents the product state |01 . . . |0N, and
|0,N represents the product state |11 . . . |1N.
FIG. 2B illustrates use of a NOON state to determine a phase shift induced by the optical medium 226. Each state in |N,0 represents a photon transmitted in a path 230, and each state in |0,N represents a photon transmitted in a path 232. Each photon in the state |0,N passes through the optical medium 226 and acquires a phase shift φ, which results in the state:
                ψ      N        〉    =            1              2              ⁢          (                                              N            ,            0                    〉                +                              ⅇ                          ⅈ              ⁢                                                          ⁢              N              ⁢                                                          ⁢              φ                                ⁢                                                0              ,              N                        〉                              )      where eiNφ represents a relative phase shift difference between the photons in the state |N,0 and photons in the state |0,N. The paths 230 and 232 intersect at a point on a substrate 234, which produces a detectable interference pattern of the form cos(Nφ) along the top surface of the substrate 234. The uncertainty associated with using the NOON state to determine the phase shift φ is, in accordance with estimation theory, given by:
      Δ    ⁢                  ⁢          φ      HL        =      1    N  
The uncertainty ΔφHL is called the “Heisenberg limit” and is the optimal accuracy obtainable for an N-trial determination of the phase shift φ. In other words, by employing N-qubit systems entangled in the NOON state, the uncertainty associated with the experimentally determined value of the phase shift φ is minimized, which is an improvement over the shot noise limit by a factor of √{square root over (N)} for the same number of trials N. However, existing methods for determining a phase shift φ using a NOON state typically employ gates and/or qubit interactions that are often cumbersome and complicated to implement. Physicists, metrologists, cryptographers, computer scientists, and quantum-information users have recognized a need for new methods that can be used to determine a phase shift φ using NOON states.